Ultrasensitive probing of plasmonic hot electron occupancies

Non-thermal and thermal carrier populations in plasmonic systems raised significant interest in contemporary fundamental and applied physics. Although the theoretical description predicts not only the energies but also the location of the generated carriers, the experimental justification of these theories is still lacking. Here, we demonstrate experimentally that upon the optical excitation of surface plasmon polaritons, a non-thermal electron population appears in the topmost domain of the plasmonic film directly coupled to the local fields. The applied all-optical method is based on spectroscopic ellipsometric determination of the dielectric function, allowing us to obtain in-depth information on surface plasmon induced changes of the directly related electron occupancies. The ultrahigh sensitivity of our method allows us to capture the signatures of changes induced by electron-electron scattering processes with ultrafast decay times. These experiments shed light on the build-up of plasmonic hot electron population in nanoscale media.


Spectra of the measured ellipsometric angles
Ellipsometric angles are defined as the amplitude ratio (Ψ) and the phase difference (Δ) between the two linearly polarized components (p and s) of polarized light illuminating the sample at nonnormal incidence (p component is parallel with the plane of incidence and s component is perpendicular to it). When applying different laser powers for exciting surface plasmon polaritons (SPPs), typical Ψ and Δ spectra could be recorded for gold films (Fig. S1), which varied with the applied laser power only slightly.

Details of the ellipsometric modeling of the plasmonic gold film
As a first step, the room-temperature (RT) data without laser excitation was analyzed.
We described the dielectric function of RT gold film with a spline based optical model [1]. The B-Spline model describes 2 by first defining evenly spaced photon energies, which are used as control points (knots or nodes) and by interpolating the optical constants between these control points. The 1 curve is calculated from the 2 curves with the help of Kramers-Kronig integration.
In our case, the spacing between the nodes was set to be 0.15 eV. To allow for Kramers-Kronig integration, two further nodes were defined outside the measured photon energy range at a 'distance' of 0.5 eV from the lower and the upper limit. To handle the effect absorption outside the investigated measurement range the amplitude of a zero oscillator located at low photon energies was fitted along with the integration constant. Using this dispersion law, and using a five-component model, namely BK7 glass, Cr adhesion layer, gold layer, surface roughness (determined by atomic force microscopy) and air, the thickness of Cr adhesion layer and gold layer could be determined. The modeling also provided the dielectric function of our gold film at room temperature. The resulting ε2 curve is plotted in Fig. S2 in comparison with literature data [2][3][4][5][6].
During the evaluation of the spectra recorded at higher temperature (100°C), the thickness values were kept fixed and only the dielectric function was fitted to deduce the optical data. This way we could set up a tabulated dielectric function dataset characterizing the temperature dependence of our gold layer. Later to determine the dielectric function of our gold film at any arbitrary temperature linear inter/extrapolation of this dataset was used assuming linear changes in the dielectric function in this temperature regime as supported by [7]. This model allowed us to set the temperature of the gold layer as fitting parameter.
In the model describing the case when SPPs are present, we divided the gold film into two sublayers, a lower layer having the temperature dependent optical model as described above (thermalized layer), and an upper layer accounting for the appearance of SPPs (non-thermalized layer). The optical properties of the non-thermalized layer were described using again the spline based optical model. The total thickness of the layer system was kept fixed. To determine the thickness of the non-thermalized layer, several different thickness values were set during the fitting procedure, during which the optical properties of the non-thermalized layers were fitted along with the temperature of the underlying thermalized layer, and the fit quality (mean squared error of the measured and fitted curves -MSE) was determined [8]. For all excitation powers, an optimal thickness could be identified, where the preset thickness resulted in a minimal MSE (Fig.   S3). In addition to the physical considerations, the existence of such minima validates the application of the two-layer model.

Comparison with topside illumination
To further prove that the non-thermalized layer exists only in the case of plasmon excitation, we performed a third type of measurement, where the gold film was illuminated from topside at normal incidence (SPP excitation is excluded). The difference curves in this case are compared to those measured upon uniform heating and upon SPP excitation of the sample (Fig. S4 -for comparison the measured difference curves are normalized.) The first two cases do not promote SPP excitation, their normalized difference curves behave similarly, and the observed changes can be attributed to the temperature rise within the sample.

Temperature of the gold film upon SPP excitation
The temperature rise of the sample upon SPP excitation was estimated additionally using two independent methods: a measurement based on reflectivity change of the gold and a simulation tool. We discuss these methods in the following sections.

Performing thermoreflectivity-based temperature estimation with ellipsometry
A broadly applied method for measuring the temperature of metal thin films is based on the measurement of the temperature dependent changes in the VIS reflectivity of metals according to the following formula [9]: where ΔR is the reflectivity change of the sample between two states exhibiting a temperature difference of ΔT, and C is a constant.
To determine the relative reflectivity increment from the recorded ellipsometric data we can directly use the recorded average intensities. First, the constant C belonging to our samples was determined from ellipsometric measurements, during which the samples were heated to preset temperatures. Then the deduced value of C was applied to evaluate the temperature of the samples during SPP excitation by using the S1 expression. To validate our ellipsometric approach, we compared the deduced C parameter with literature values [9][10][11][12] and found very good agreement (Fig. S5).

Temperature calculations
To support that the effect of SPP excitation is not a thermal gradient within the layer but the measured data can only be interpreted as an additional electron population with a non-Fermi-Dirac energy distribution, we carried out thermal modeling using COMSOL Multiphysics. For the calculations, we considered that the laser power that is spent on plasmon excitation is absorbed in the top boundary of the gold layer (gold-air interface). For this in our SPP excitation setup, we measured the power of the incoming laser beam and the power of the reflected beam both in sand p-polarization states. The difference between these two polarizations will account for surface plasmon excitation, since SPPs are generated only in p-polarization state, while with the comparison with s-polarization we can take into account the possible losses due to e.g. reflections on the different interfaces. In short, with this estimation, we assume that all the power -that is not reflected -will excite plasmons, and all the power of this plasmon excitation will then be converted to the heating of the gold.
In our simulation, this absorbed power is fed to a heat source located at the top 5 nm of the gold layer. With such location of the heat source, we tried to mimic the field localization property of the SPP generation, namely that a significant part of the electromagnetic energy of the incoming field is transformed to the kinetic energy of free carriers, which are confined within a few-nm vicinity of the surface (due to Thomas-Fermi screening or Friedel-oscillations).
The modeled part of our investigated system contains a 10 mm thick glass substrate, 45 nm gold layer on the top and the surrounding air. The simulation domain is 3D with cylindrical symmetry.
The modeled cylindrical volume has 4 mm diameter with 260-μm diameter heat source representing the laser spot.
The resulting steady-state temperature distribution shows negligible difference between the top and the bottom part of the gold layer (∆T<0.1 K) supporting that the measured signal is not the consequence of a temperature gradient. Furthermore, the calculated temperatures of the gold layer for the different applied laser power values coincide with the ones deduced from the reflection based and from the ellipsometric measurements within the error bars (see Fig. 3 of the main article).

Effect of SPP electric field on the optical properties
To reveal the possibility of nonlinear effects on the optical properties, we considered the maximal intensity on the surface of the gold film taking into account also the plasmonic field enhancement effect, and calculated the nonlinear refractive index using the available nonlinear optical parameters from the literature.
In our excitation geometry, the intensity is 131.25 W/cm 2 for the largest applied laser power. We calculated the electric field distribution in the direction perpendicular to the sample surface using Lumerical FDTD Solutions software under the same conditions applied during our experiments.
The nonlinear refractive index can be estimated with the following formula: n=n1+n2*I, where n1 is the linear refractive index, and n2 is the term characterizing the nonlinearity in m 2 /W units. I is the intensity. Since the literature values of the n2 for bulk gold range from 10 -12 to 10 -16 m 2 /W [13], the nonlinear term has a negligible -at least 4 orders of magnitude smaller than the linear term -contribution in our case. Moreover, since the field strength inside the gold film drops significantly and decays within a few tens of nanometers, the effect is expected to be even smaller. Fig. S6. Local intensity in our excitation geometry inside the gold film and in its vicinity considering also the field enhancement property of the generated SPPs.

Excluding possible nonlocal effects
One would expect that the observed changes can be attributed to nonlocal effects. However, according to [14,15] nonlocality manifests in anisotropic behavior of the dielectric function. To test the possibility of nonlocal effects, we carried out our measurements when the plane of the ellipsometric setup is parallel or perpendicular with respect to the propagation direction of the SPPs. The measured ellipsometric difference curves show no deviations depending on the excitation geometry (Fig. S7). This supports that no anisotropic effects have to be taken into account.

Parameters for EDJDOS calculation
For calculating the EDJDOS, parabolic band structures were assumed at the L and X points in the Brillouin zone with the following band energies and masses (sources included). With these parameters the interband contributions and resulting 2 curve is shown in Fig. S8.  Upon the evaluation of the retrieved Δε2 data we deduced the Δf(E) curve with a simple spline based fitting method. In this method, first we define energies where the actual value of the Δf(E) curve can be adjusted (knots or nodes). Second, between these energy values the Δf(E) curve is interpolated with the help of a quadratic spline. As a third step, we adjust the Δf(E) values at the knots with a fitting algorithm, to describe the previously retrieved Δ curves in the 1.8-2.7 eV range.

Comparison with thermal effects
To analyse how the retrieved Δf(E) curves relate to Δf(E) curves belonging to pure thermal effects, we analyzed purely thermal Δε2 curves calculated from literature data [5] (providing access to datasets measured at higher temperatures) by using two Fermi-Dirac distributions. The direct comparison of thermal and non-thermal (plasmon-assisted) Δε2 curves shows already rather important differences: the shape of the 'baseline' is rather different indicating a different change in the free electron properties, i.e. Drude terms/intraband contribution. Furthermore, the shape of the main peak at around 2.3 eV is wider for the non-thermal case. For the purely thermal case, the base temperature was set to the lattice temperature belonging to the highest laser intensity of the plasmon excitation case (105 mW). The temperature difference was set so that the amplitude of the Δf(E) curves retrieved at 105 mW coincide with that calculated from the two Fermi-Dirac curves. From the point of view of the reliability of our method, the purely temperature dependent reference Δε2 data could be well described with the help of the Fermi-Dirac distributions when taking into account a slight shift in the band structure for the higher lattice temperature due to lattice expansion during heating [16]. Regarding the shape of the Δf(E) curves, the extent of the broadening is much larger for the highest laser intensity than that of a thermal system, holding the spectral fingerprint of an additional hot electron population with energy levels up to 0.4 eV measured from the Fermi level (Fig. S9).